The H-trivial line-bundle and Picard-region equivalence for toric DM stacks

For $m\in \mathbb{N}$, let $\mathbb{P}_{\mathbf{\Sigma}}$ be a proper smooth dimension $m$ toric DM stack associated to a complete stacky fan $\mathbf{\Sigma}=(\Sigma, \{v_i\}_{i=1}^{n})$. Let $\operatorname{Pic}(\mathbb{P}_{\mathbf{\Sigma}})$ be its Picard group, let $\Delta$ be the index set used in the paper, and let $Z_I$ be the corresponding regions. An H-trivial line bundle means a line bundle satisfying the paper's H-triviality condition.

The H-trivial line-bundle and Picard-region conjecture. There are infinitely many H-trivial line bundles on $\mathbb{P}_{\mathbf{\Sigma}}$ if and only if there exists a nonzero element $\mathcal{L}\in \operatorname{Pic}(\mathbb{P}_{\mathbf{\Sigma}})$ which is not contained in the interior of $Z_I$ for any $I\in\Delta$.

The paper presents this as a weaker conjecture: it asserts the equivalence of statements (2) and (3) from the surrounding discussion. The implication from infinitely many H-trivial line bundles to the Picard-region condition is stated to follow in full generality from the cited argument, while the converse remains open in general.